Testing One Hypothesis Multiple Times: The Multidimensional Case

Abstract

The identification of new rare signals in data, the detection of a sudden change in a trend, and the selection of competing models are among the most challenging problems in statistical practice. These challenges can be tackled using a test of hypothesis where a nuisance parameter is present only under the alternative, and a computationally efficient solution can be obtained by the “testing one hypothesis multiple times” (TOHM) method. In the one-dimensional setting, a fine discretization of the space of the non identifiable parameter is specified, and a global p-value is obtained by approximating the distribution of the supremum of the resulting stochastic process. In this article, we propose a computationally efficient inferential tool to perform TOHM in the multidimensional setting. Here, the approximations of interest typically involve the expected Euler characteristics (EC) of the excursion set of the underlying random field. We introduce a simple algorithm to compute the EC in multiple dimensions and for arbitrarily large significance levels. This leads to an highly generalizable computational tool to perform hypothesis testing under nonstandard regularity conditions. Supplementary materials for this article are available online.